Now we know that sounds are at the same pitch when they are generated by sonorous bodies having the same speed of vibration, and it is easy to perceive that, if two such bodies are sounding together, the condensations and rarefactions of the layers of air will synchronize with each other, so that both will be exciting condensations at the same instant and likewise will generate rarefactions simultaneously. And even if the two bodies have not exactly the same speed, the result will be equally simple as long as their speeds bear simple ratios to each other. Thus two bodies which are emitting sounds at the interval of an octave or of a fifth or fourth will generate condensations and rarefactions in such a manner that they will not interfere one with another. But the case is different where two sounds are separated by differences in pitch that cannot be expressed by simple ratios. For example, if one sound be one vibration per second higher than another, it is clear that by the time that the first sounding body has completed its given number of vibrations in one second, the other will be one vibration behind. When, therefore, the vibrations of the first body are continued into the next second the condensation of one wave will be completely synchronous with neither the condensation nor the rarefaction of the other. The obvious result is that at a certain point the condensations of each wave concur while at another point the condensation of one crosses the rarefaction of the other. In the first case we have a considerable augmentation of sound and in the other case a complete silence. As the waves thus approach and recede there is a gradual diminution of sound followed by a complete cessation for a small fraction of a second, and then a gradual increase until the point of greatest augmentation occurs. This latter happens when the two condensations concur, and the gradual rise and fall of the sound correspond to the gradual approach of this concurrence in the first case and to the similar advance of the point of crossing in the second. This phenomenon of alternate augmentation and diminution of sound separated by an almost inappreciable interval of silence occurs whenever two sounds of nearly the same pitch are heard simultaneously. These peculiar changes in the intensity of a sound are denominated “beats.”